Functional Analysis HW #1
نویسنده
چکیده
• Suppose that |X | = ∞. Given any n ∈ Z, choose a set of n distinct points Xn = {x1, · · · , xn } ⊂ X . We claim that Claim 1.1. The mapping α : C(X ) → C(Xn ) given by α( f ) = f |Xn is a surjective linear map. Assuming this, we know that dim C(X ) ≥ C(Xn ) = n for any n. Therefore dim C(X ) = ∞. To finish the proof we are required to show Claim 1.1. It is straightforward that α is a well-defined linear map, thus it suffices to prove that α is surjective. Since X is compact Hausdorff, it is a normal space. Then by the Urysohn’s lemma, for each i, j ∈ {1, · · · , n} with i , j we can choose f i j ∈ C(X ) such that f i j (xi ) = 1 and f i j (x j ) = 0. Now define gi = ∏ j : j,i f j . Then gi ∈ C(X ) and gi (x j ) = δi j . This proves that α is surjective and hence the claim.
منابع مشابه
Functional Analysis HW
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